Kaprekar's routine

In number theory, Kaprekar's routine is an iterative algorithm that, with each iteration, takes a natural number in a given number base, creates two new numbers by sorting the digits of its number by descending and ascending order, and subtracts the second from the first to yield the natural number for the next iteration. It is named after its inventor, the Indian mathematician D. R. Kaprekar.

Kaprekar showed that in the case of 4-digit numbers in base 10, if the initial number has at least two distinct digits, after 7 iterations this process always yields the number 6174, which is now known as Kaprekar's constant.^[1]

Definition and properties[]

The algorithm is as follows:^[2]

Choose any natural number $n$ in a given number base $b$ . This is the first number of the sequence.
Create a new number $\alpha$ by sorting the digits of $n$ in descending order, and another new number $\beta$ by sorting the digits of $n$ in ascending order. These numbers may have leading zeros, which are discarded (or alternatively, retained). Subtract $\alpha -\beta$ to produce the next number of the sequence.
Repeat step 2.

The sequence is called a Kaprekar sequence and the function $K_{b}(n)=\alpha -\beta$ is the Kaprekar mapping. Some numbers map to themselves; these are the fixed points of the Kaprekar mapping,^[3] and are called Kaprekar's constants. Zero is a Kaprekar's constant for all bases $b$ , and so is called a trivial Kaprekar's constant. All other Kaprekar's constant are nontrivial Kaprekar's constants.

For example, in base 10, starting with 3524,

K_{10}(3524)=5432-2345=3087

K_{10}(3087)=8730-378=8352

K_{10}(8352)=8532-2358=6174

K_{10}(6174)=7641-1467=6174

with 6174 as a Kaprekar's constant.

All Kaprekar sequences will either reach one of these fixed points or will result in a repeating cycle. Either way, the end result is reached in a fairly small number of steps.

Note that the numbers $\alpha$ and $\beta$ have the same digit sum and hence the same remainder modulo $b-1$ . Therefore, each number in a Kaprekar sequence of base $b$ numbers (other than possibly the first) is a multiple of $b-1$ .

When leading zeroes are retained, only repdigits lead to the trivial Kaprekar's constant.

Families of Kaprekar's constants[]

In base 4, it can easily be shown that all numbers of the form 3021, 310221, 31102221, 3...111...02...222...1 (where the length of the "1" sequence and the length of the "2" sequence are the same) are fixed points of the Kaprekar mapping.

In base 10, it can easily be shown that all numbers of the form 6174, 631764, 63317664, 6...333...17...666...4 (where the length of the "3" sequence and the length of the "6" sequence are the same) are fixed points of the Kaprekar mapping.

b = 2k[]

It can be shown that all natural numbers

m=(k)b^{2n+3}\left(\sum _{i=0}^{n-1}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b\left(\sum _{i=0}^{n-1}b^{i}\right)+(k)

are fixed points of the Kaprekar mapping in even base $b=2k$ for all natural numbers $n$ .

Proof

$\alpha =(2k-1)b^{2n+2}\left(\sum _{i=0}^{n}b^{i}\right)+(k)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)\left(\sum _{i=0}^{n}b^{i}\right)$

$\beta =(k-1)b^{2n+2}\left(\sum _{i=0}^{n}b^{i}\right)+(k)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(2k-1)\left(\sum _{i=0}^{n}b^{i}\right)$

${\begin{aligned}K_{b}(m)&=\alpha -\beta \\&=((2k-1)-(k-1))b^{2n+2}\left(\sum _{i=0}^{n}b^{i}\right)+(k-k)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+((k-1)-(2k-1))\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+2}\left(\sum _{i=0}^{n}b^{i}\right)-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+b^{2n+2}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k)b^{2n+1}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{2n+1}+b^{2n+1}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{2n+1-1}\left(\sum _{i=0}^{1}b^{i}\right)+b^{2n+1-1}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{2n+1-n}\left(\sum _{i=0}^{n}b^{i}\right)+b^{2n+1-n}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+b^{n+1}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(2k)b^{n}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+kb^{n}-k\left(\sum _{i=0}^{n-1}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{n+1-1}+b^{n+1-1}-k\left(\sum _{i=0}^{n-n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{n+1-n}\left(\sum _{i=0}^{n}b^{i}\right)+b^{n+1-n}-k\left(\sum _{i=0}^{n-n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b\left(\sum _{i=0}^{n}b^{i}\right)+b-k\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b\left(\sum _{i=0}^{n}b^{i}\right)+2k-k\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b\left(\sum _{i=0}^{n}b^{i}\right)+k\\&=m\\\end{aligned}}$

Perfect digital invariants
$k$	$b$	$m$
1	2	011, 101101, 110111001, 111011110001...
2	4	132, 213312, 221333112, 222133331112...
3	6	253, 325523, 332555223, 333255552223...
4	8	374, 437734, 443777334, 444377773334...
5	10	495, 549945, 554999445, 555499994445...
6	12	5B6, 65BB56, 665BBB556, 6665BBBB5556...
7	14	6D7, 76DD67, 776DDD667, 7776DDDD6667...
8	16	7F8, 87FF78, 887FFF778, 8887FFFF7778...
9	18	8H9, 98HH89, 998HHH889, 9998HHHH8889...

Kaprekar's constants and cycles of the Kaprekar mapping for specific base b[]

All numbers are expressed in base $b$ , using A−Z to represent digit values 10 to 35.

Base $b$	Digit length	Nontrivial (nonzero) Kaprekar's constants	Cycles
2	2	01^{[note 1]}	$\varnothing$
	3	011^{[note 1]}	$\varnothing$
	4	0111,^{[note 1]} 1001	$\varnothing$
	5	01111,^{[note 1]} 10101	$\varnothing$
	6	011111,^{[note 1]} 101101, 110001	$\varnothing$
	7	0111111,^{[note 1]} 1011101, 1101001	$\varnothing$
	8	01111111,^{[note 1]} 10111101, 11011001, 11100001	$\varnothing$
	9	011111111,^{[note 1]} 101111101, 110111001, 111010001	$\varnothing$
3	2	$\varnothing$	$\varnothing$
	3	$\varnothing$	022 → 121 → 022^{[note 1]}
	4	$\varnothing$	1012 → 1221 → 1012
	5	20211	$\varnothing$
	6	$\varnothing$	102212 → 210111 → 122221 → 102212
	7	2202101	2022211 → 2102111 → 2022211
	8	21022111	$\varnothing$
	9	222021001	220222101 → 221021101 → 220222101 202222211 → 210222111 → 211021111 → 202222211
4	2	$\varnothing$	03 → 21 → 03^{[note 1]}
	3	132	$\varnothing$
	4	3021	1332 → 2022 → 1332
	5	$\varnothing$	20322 → 23331 → 20322
	6	213312, 310221, 330201	$\varnothing$
	7	3203211	$\varnothing$
	8	31102221, 33102201, 33302001	22033212 → 31333311 → 22133112 → 22033212
	9	221333112, 321032211, 332032101	$\varnothing$
5	2	13	$\varnothing$
	3	$\varnothing$	143 → 242 → 143
	4	3032	$\varnothing$
6	2	$\varnothing$	05 → 41 → 23 → 05^{[note 1]}
	3	253	$\varnothing$
	4	$\varnothing$	1554 → 4042 → 4132 → 3043 → 3552 → 3133 → 1554
	5	41532	31533 → 35552 → 31533
	6	325523, 420432, 530421	205544 → 525521 → 432222 → 205544
	7	$\varnothing$	4405412 → 5315321 → 4405412
	8	43155322, 55304201	31104443 → 43255222 → 33204323 → 41055442 → 54155311 → 44404112 → 43313222 → 31104443 42104432 → 43204322 → 42104432 53104421 → 53304221 → 53104421
7	2	$\varnothing$	$\varnothing$
	3	$\varnothing$	264 → 363 → 264
	4	$\varnothing$	3054 → 5052 → 5232 → 3054
8	2	25	07 → 61 → 43 → 07^{[note 1]}
	3	374	$\varnothing$
	4	$\varnothing$	1776 → 6062 → 6332 → 3774 → 4244 → 1776 3065 → 6152 → 5243 → 3065
	5	$\varnothing$	42744 → 47773 → 42744 51753 → 61752 → 63732 → 52743 → 51753
	6	437734, 640632	310665 → 651522 → 532443 → 310665
9	2	$\varnothing$	17 → 53 → 17
	3	$\varnothing$	385 → 484 → 385
	4	$\varnothing$	3076 → 7252 → 5254 → 3076 5074 → 7072 → 7432 → 5074
10^[4]	2	$\varnothing$	09 → 81 → 63 → 27 → 45 → 09^{[note 1]}
	3	495	$\varnothing$
	4	6174	$\varnothing$
	5	$\varnothing$	53955 → 59994 → 53955 61974 → 82962 → 75933 → 63954 → 61974 62964 → 71973 → 83952 → 74943 → 62964
	6	549945, 631764	420876 → 851742 → 750843 → 840852 → 860832 → 862632 → 642654 → 420876
	7	$\varnothing$	7509843 → 9529641 → 8719722 → 8649432 → 7519743 → 8429652 → 7619733 → 8439552 → 7509843
	8	63317664, 97508421	43208766 → 85317642 → 75308643 → 84308652 → 86308632 → 86326632 → 64326654 → 43208766 64308654 → 83208762 → 86526432 → 64308654
11	2	37	$\varnothing$
	3	$\varnothing$	4A6 → 5A5 → 4A6
	4	$\varnothing$	3098 → 9452 → 7094 → 9272 → 7454 → 3098 5096 → 9092 → 9632 → 7274 → 5276 → 5096
12	2	$\varnothing$	0B → A1 → 83 → 47 → 29 → 65 → 0B^{[note 1]}
	3	5B6	$\varnothing$
	4	$\varnothing$	3BB8 → 8284 → 6376 → 3BB8 4198 → 8374 → 5287 → 6196 → 7BB4 → 7375 → 4198
	5	83B74	64B66 → 6BBB5 → 64B66
	6	65BB56	420A98 → A73742 → 842874 → 642876 → 62BB86 → 951963 → 860A54 → A40A72 → A82832 → 864654 → 420A98
	7	962B853	841B974 → A53B762 → 971B943 → A64B652 → 960BA53 → B73B741 → A82B832 → 984B633 → 863B754 → 841B974
	8	873BB744, A850A632	4210AA98 → A9737422 → 87428744 → 64328876 → 652BB866 → 961BB953 → A8428732 → 86528654 → 6410AA76 → A92BB822 → 9980A323 → A7646542 → 8320A984 → A7537642 → 8430A874 → A5428762 → 8630A854 → A540X762 → A830A832 → A8546632 → 8520A964 → A740A742 → A8328832 → 86546654
13	2	$\varnothing$	1B → 93 → 57 → 1B
13	3	$\varnothing$	5C7 → 6C6 → 5C7
14	2	49	2B → 85 → 2B 0D → C1 → A3 → 67 → 0D^{[note 1]}
14	3	6D7	$\varnothing$
15	2	$\varnothing$	$\varnothing$
15	3	$\varnothing$	6E8 → 7E7 → 6E8
16^[5]	2	$\varnothing$	2D → A5 → 4B → 69 → 2D 0F → E1 → C3 → 87 → 0F^{[note 1]}
	3	7F8	$\varnothing$
	4	$\varnothing$	3FFC → C2C4 → A776 → 3FFC A596 → 52CB → A596 E0E2 → EB32 → C774 → 7FF8 → 8688 → 1FFE → E0E2 E952 → C3B4 → 9687 → 30ED → E952
	5	$\varnothing$	86F88 → 8FFF7 → 86F88 A3FB6 → C4FA4 → B7F75 → A3FB6 A4FA6 → B3FB5 → C5F94 → B6F85 → A4FA6
	6	87FF78	310EED → ED9522 → CB3B44 → 976887 → 310EED 532CCB → A95966 → 532CCB 840EB8 → E6FF82 → D95963 → A42CB6 → A73B86 → 840EB8 A80E76 → E40EB2 → EC6832 → C91D64 → C82C74 → A80E76 C60E94 → E82C72 → CA0E54 → E84A72 → C60E94
	7	C83FB74	B62FC95 → D74FA83 → C92FC64 → D85F973 → C81FD74 → E94FA62 → DA3FB53 → CA5F954 → B74FA85 → B62FC95 B71FD85 → E83FB72 → DB3FB43 → CA6F854 → B73FB85 → C63FB94 → C84FA74 → B82FC75 → D73FB83 → CA3FB54 → C85F974 → B71FD85
	8	$\varnothing$	3110EEED → EDD95222 → CBB3B444 → 97768887 → 3110EEED 5332CCCB → A9959666 → 5332CCCB 7530ECA9 → E951DA62 → DB52CA43 → B974A865 → 7530ECA9 A832CC76 → A940EB66 → E742CB82 → CA70E854 → E850EA72 → EC50EA32 → EC94A632 → C962C964 → A832CC76 C610EE94 → ED82C722 → CBA0E544 → E874A872 → C610EE94 C630EC94 → E982C762 → CA30EC54 → E984A762 → C630EC94 C650EA94 → E852CA72 → CA50EA54 → E854AA72 → C650EA94 CA10EE54 → ED84A722 → CB60E944 → E872C872 → CA10EE54

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p Leading zeroes retained.

Kaprekar's constants in base 10[]

Numbers of length four digits[]

In 1949 D. R. Kaprekar discovered^[6] that if the above process is applied to base 10 numbers of 4 digits, the resulting sequence will almost always converge to the value 6174 in at most 8 iterations, except for a small set of initial numbers which converge instead to 0. The number 6174 is the first Kaprekar's constant to be discovered, and thus is sometimes known as Kaprekar's constant.^[7]^[8]^[9]

The set of numbers that converge to zero depends on whether leading zeros are retained (the usual formulation) or are discarded (as in Kaprekar's original formulation).

In the usual formulation, there are 77 four-digit numbers that converge to zero,^[10] for example 2111. However, in Kaprekar's original formulation the leading zeros are retained, and only repdigits such as 1111 or 2222 map to zero. This contrast is illustrated below:

discard leading zeros	retain leading zeros
2111 − 1112 = 999 999 − 999 = 0	2111 − 1112 = 0999 9990 − 0999 = 8991 9981 − 1899 = 8082 8820 − 0288 = 8532 8532 − 2358 = 6174

Below is a flowchart. Leading zeros are retained, however the only difference when leading zeros are discarded is that instead of 0999 connecting to 8991, we get 999 connecting to 0.

Sequence of Kaprekar transformations ending in 6174

Numbers of length three digits[]

If the Kaprekar routine is applied to numbers of 3 digits in base 10, the resulting sequence will almost always converge to the value 495 in at most 6 iterations, except for a small set of initial numbers which converge instead to 0.^[7]

The set of numbers that converge to zero depends on whether leading zeros are discarded (the usual formulation) or are retained (as in Kaprekar's original formulation). In the usual formulation, there are 60 three-digit numbers that converge to zero,^[11] for example 211. However, in Kaprekar's original formulation the leading zeros are retained, and only repdigits such as 111 or 222 map to zero.

Below is a flowchart. Leading zeros are retained, however the only difference when leading zeros are discarded is that instead of 099 connecting to 891, we get 99 connecting to 0.

Sequence of three digit Kaprekar transformations ending in 495

Other digit lengths[]

For digit lengths other than three or four (in base 10), the routine may terminate at one of several fixed points or may enter one of several cycles instead, depending on the starting value of the sequence.^[7] See the table in the section above for base 10 fixed points and cycles.

The number of cycles increases rapidly with larger digit lengths, and all but a small handful of these cycles are of length three. For example, for 20-digit numbers in base 10, there are fourteen constants (cycles of length one) and ninety-six cycles of length greater than one, all but two of which are of length three. Odd digit lengths produce fewer different end results than even digit lengths.^[12]^[13]

Programming example[]

The example below implements the Kaprekar mapping described in the definition above to search for Kaprekar's constants and cycles in Python.

Leading zeroes discarded[]

def get_digits(x, b):
    digits = []
    while x > 0:
        digits.append(x % b)
        x = x // b
    return digits
    
def form_number(digits, b):
    result = 0
    for i in range(0, len(digits)):
        result = result * b + digits[i]
    return result

def kaprekar_map(x, b):
    descending = form_number(sorted(get_digits(x, b), reverse=True), b)
    ascending = form_number(sorted(get_digits(x, b)), b)
    return descending - ascending
    
def kaprekar_cycle(x, b):
    x = int (str(x), b)
    seen = []
    while x not in seen:
        seen.append(x)
        x = kaprekar_map(x, b)
    cycle = []
    while x not in cycle:
        cycle.append(x)
        x = kaprekar_map(x, b)
    return cycle

Leading zeroes retained[]

def digit_count(x, b):
    count = 0
    while x > 0:
        count = count + 1
        x = x // b
    return count
    
def get_digits(x, b, init_k):
    k = digit_count(x, b)
    digits = []
    while x > 0:
        digits.append(x % b)
        x = x // b
    for i in range(k, init_k):
        digits.append(0)
    return digits
    
def form_number(digits, b):
    result = 0
    for i in range(0, len(digits)):
        result = result * b + digits[i]
    return result
    
def kaprekar_map(x, b, init_k):
    descending = form_number(sorted(get_digits(x, b, init_k), reverse=True), b)
    ascending = form_number(sorted(get_digits(x, b, init_k)), b)
    return descending - ascending
    
def kaprekar_cycle(x, b):
    x = int (str(x), b)
    init_k = digit_count(x, b)
    seen = []
    while x not in seen:
        seen.append(x)
        x = kaprekar_map(x, b, init_k)
    cycle = []
    while x not in cycle:
        cycle.append(x)
        x = kaprekar_map(x, b, init_k)
    return cycle

Citations[]

^ Hanover 2017, p. 1, Overview.
^ Hanover 2017, p. 3, Methodology.
^ (sequence A099009 in the OEIS)
^ "Sample Kaprekar Series".
^ "Sample Kaprekar Series for hexadecimal numbers".
^ Kaprekar DR (1955). "An Interesting Property of the Number 6174". Scripta Mathematica. 15: 244–245.
^ ^a ^b ^c Weisstein, Eric W. "Kaprekar Routine". MathWorld.
^ Yutaka Nishiyama, Mysterious number 6174
^ Kaprekar DR (1980). "On Kaprekar Numbers". Journal of Recreational Mathematics. 13 (2): 81–82.
^ (sequence A069746 in the OEIS)
^ (sequence A090429 in the OEIS)
^ "Sample Kaprekar Series".
^ "Playing with Numbers".

References[]

Hanover, Daniel (2017). "The Base Dependent Behavior of Kaprekar's Routine: A Theoretical and Computational Study Revealing New Regularities". International Journal of Pure and Applied Mathematics. arXiv:1710.06308.

External links[]

Bowley, Rover. "6174 is Kaprekar's Constant". Numberphile. University of Nottingham: Brady Haran. Archived from the original on 2017-08-23. Retrieved 2013-04-01.
Sample (Perl) code to walk any four-digit number to Kaprekar's Constant

[retained-4] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p Leading zeroes retained.

[FOOTNOTEHanover20171Overview-1] Hanover 2017, p. 1, Overview.

[FOOTNOTEHanover20173Methodology-2] Hanover 2017, p. 3, Methodology.

[A099009-3] (sequence A099009 in the OEIS)

[sourceforge_dec-5] "Sample Kaprekar Series".

[sourceforge_hex-6] "Sample Kaprekar Series for hexadecimal numbers".

[Kaprekar1955-7] Kaprekar DR (1955). "An Interesting Property of the Number 6174". Scripta Mathematica. 15: 244–245.

[mathworld-8] Weisstein, Eric W. "Kaprekar Routine". MathWorld.

[9] Yutaka Nishiyama, Mysterious number 6174

[Kaprekar1980-10] Kaprekar DR (1980). "On Kaprekar Numbers". Journal of Recreational Mathematics. 13 (2): 81–82.

[11] (sequence A069746 in the OEIS)

[12] (sequence A090429 in the OEIS)

[13] "Sample Kaprekar Series".

[14] "Playing with Numbers".

[1]

[2]

[3]

[note 1]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]